Square Root of 222

√222 at a glance

Exact value

√222

Decimal (10 places)

14.8996644258

Rounded

14.9 · 14.90 · 14.900

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.899664

Prime factorization

2 × 3 × 37

Cube root

6.055049

How to simplify √222

The prime factorization of 222 is 2 × 3 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √222 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 222, 2, 3 and 37 appear an odd number of times, so √222 is irrational and 14.8996644258 is a rounded value.

Where √222 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √222 lies between 14 and 15. 222 is 26 above 196 and 3 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.8966 (0.02% low)
• Tangent from 14, i.e. 14 + 26 ÷ 28: 14.9286 (0.19% high)
• Tangent from 15, i.e. 15 − 3 ÷ 30: 14.9000 (0% high)

For √222 the tangent at 15 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 222 is just 3 below 225.

Finding √222 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 15 (15² = 225):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √222 = 14.8996644258 to every decimal shown.

√222 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √222 the pattern is [14; 1, 8, 1, 28] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √222 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 222y² = 1. Its smallest solution in positive whole numbers is x = 149, y = 10.

√222 in geometry and everyday measurements

• A square patio or deck of 222 square feet is about 14.9 ft (14 ft 11 in) on each side, so edging all the way around takes 4 × √222 ≈ 59.6 ft.
• 222 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √222 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 14 box, because 1² + 5² + 14² = 222.

Square roots near √222 and related roots

• The cube root of 222 is about 6.055049.
• Four times the radicand doubles the root: √888 = 2 × √222 ≈ 29.799329.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.